Note the cells a, b, c. Either c is <4>, or the XY-wing 45, 24, 25 is true.

Asellus made the rather brilliant observation that if the XY-wing is true, c is solved as <5> (look at C5). Either way, c cannot be <2>. This solves the puzzle.

In the grid above, there is also an XYZ-wing, a, d, e. Note that you can regard an XYZ-wing as a finned XY-wing. In this case, either the pivot cell, d, is <4>, or the XY-wing 45, 25, 24 is true. Either way, R3C7 is not <4>. This XYZ-wing does not solve the puzzle.

Keith

Last edited by keith on Sun Jul 27, 2008 12:04 pm; edited 1 time in total

The finned XY-wing is just a way to start a bifurcation: Either "A" is true, or "B" is true. Presumably, you could just follow A and B until you find a candidate that is either true, or one that is false, for both A and B.

But, the hope is that this is not just another way to start a trial-and-error elimination. As Asellus might say, "Are there compact patterns we can look for?"

I think so. Suppose the finned wing is:

XZ-XY-VYZ

Either the XY-wing (XZ-XY-YZ) is true, or the fin (V) is true.

1. Suppose the XY-wing solves the finned cell as Z. Then you can eliminate Y as a candidate in the finned cell. That is Asellus' elimination in the first post of this thread. (You can interchange Y and Z here. It does not matter which wing candidate is solved in the finned cell.)

2. Suppose the XY-wing solves some other cell as V. Then, that solved cell and the finned cell are a strong link. V can be eliminated in all cells that see them both. I think ravel and I have both used this, see the second message of the thread, and the links in the third.

3. Note that 2. above is a true strong link. In the solution, only one of the cells is V. So, the link can be put in any chain in place of a "regular" strong link. I think it was Asellus who noted that the stong links revealed by finned XY-wings can otherwise be very difficult to recognize.

4. It may be that both the XY-wing and the fin lead to some other candidate, say "U", being true in two different cells. Again, those two cells are pincers, eliminating U in any cell that sees them both.

But, I think this last one is outside the spirit of trying to find "compact" eliminations for those of use who use pencil and paper.

Keith

(I am now done for the time being. No more posts from me until others post responses on this thread.)

Last edited by keith on Mon Jul 28, 2008 12:04 am; edited 2 times in total

3. Note that 2. above is a true strong link. In the solution, only one of the cells is V. So, the link can be put in any chain in place of a "regular" strong link. I think it was Asellus who noted that the stong links revealed by finned XY-wings can otherwise be very difficult to recognize.

Yes, I think it probably was.

I would hasten to add that anything determined to be true IF the potential XY Wing is true is thus strongly linked with the fin digit(s). [Note that the fin doesn't necessarily have to be a single digit. However, using a multi-digit fin can be tricky and there are no examples of such a thing posted as yet.] It is important to note, however, that these strong links are strong inference links and not necessarily conjugate (strong) links. So, be sure you understand this distinction if you are going to employ these links in chains.

On a related note, in my original posts on this subject, I cautiously stated that the fin-to-wing link was a "unidirectional" strong link. I have now decided that this was not correct and have posted an explanatory edit to the original post. So... you can forget about that "unidirectional" stuff!

I am not sure I understand the distinction between "Inferential" and "Conjugate", but that is OK. I am not a programmer (any more), and I check my chains in both directions to make sure they are correct.

But, I do have a question. Suppose the fin is V, and the XY-wing solves some other cell as U. Can I do Medusa coloring on U and the fin, V, as if it was a two-candidate cell?

Suppose my Medusa colors are Red and Green.

If the fin is Red, can I color U in its cell as Green? (I think so.)

If U in its cell is Green, can I color the fin Red? (I am less sure.)

Maybe I am questioning your directional assertion. It is certainly possible that the XY-wing is true and the fin is true both lead to some cell where T is true. See my original post. If T is true, can I make any statement about the the finned XY-wing?

Keith

[The heck with Holiday Inn Express. The tag line should be: "... but I did have a product of Sonoma County last night."]

But, I do have a question. Suppose the fin is V, and the XY-wing solves some other cell as U. Can I do Medusa coloring on U and the fin, V, as if it was a two-candidate cell?

In general, no. This gets to that strong inference versus conjugate distinction (and for me, it is all about the logic of sudoku solving and nothing whatsoever about computer programming!). A strong inference exists between two things when they cannot both be false. That is the case here. U and V cannot both be false so they possess a strong (inference) link. But, they are not necessarily conjugate: is it possible that both of them might be true.

(While people commonly call conjugate links "strong" links, it is actually the case that they are both strong... "cannot both be false"... and weak ... "cannot both be true". That is why a conjugate link can "substitute" for a weak link in an AIC. It isn't really substituting for a weak link because it IS a weak link, as well as being a strong link. But, our U and V link is only strong as far as we know.)

However, there is a way to exploit this U=V link with Medusa. But, you must use multi-coloring. Let's say we color U "A". Then, we use this to color an "Aa" cluster. Next, we color V "B" and construct a "Bb" cluster from that. Now, all "A"s are strongly linked with all "B"s and so can perform trapping. However, the "a"s and the "b"s are weakly linked and so cannot trap anything.

Usually, folks "bridge" two color clusters by exploiting a weak link between them, in which case the strong "trapping" colors are the ones opposite those that are involved in the weak link bridge. But, a "strong only" inference link can also bridge two clusters, as in this case. The difference is that, now, the strong "trapping" colors are the ones involved in the bridge.

Quote:

Maybe I am questioning your directional assertion.

The bidirectional nature of the strong inference is a separate issue. Perhaps you are uncertain because you are trying to see this as a conjugate link, which it is not. As a strong inference, it is bidirectional: V-false implies U-true and U-false implies V-true. But, V-true doesn't imply anything; nor does U-true.

Quote:

It is certainly possible that the XY-wing is true and the fin is true both lead to some cell where T is true. See my original post. If T is true, can I make any statement about the the finned XY-wing?

There is no "if" about it: T must be true! Other than saying that the Finned XY Wing makes T true, what would you like to say about it? This is a different situation from what we were considering above. Here, you have a discontinous AIC loop were T is connected by strong links on both ends. We could represent this using a generic chain notation, to which I will add the true-false implication sequences written below it:

Code:

T = a - V = Wing - b = T

L to R --> F T F T F T
T F T F T F <-- R to L

(AICs of any length may be inserted in the "a" and "b" positions. As before, V is the fin.) No matter which way we read the chain, an assumption that T is false leads to a contradiction (that T must be true). So, T must be true.

Now... in the case that all of the links in this loop other than "V=Wing" are conjugate links, then both the fin and the wing must be true! This might strike you as impossible. But, it isn't. That's because the statement "the XY Wing is true" for these implication purposes means only that at least one of its pincer digits is true (since that is all that is required to render the wing victims false). And, that can happen even if the "ZX-XY-YZ" pattern itself is destroyed, as it is if the fin is true.

I hope this has answered your questions... assuming you're still with me.

This puzzle is from "Die Presse" Saturday 2nd of August and was posted by nataraj in the other puzzles forum, the puzzle can be solved with the advanced VH techniques.

Reading the latest posts by Keith, Asellus and others, for finned xy-wings i tried to spot one of these in the grid, but found another(I'm not sure about this) finned creature, a finned xyz-wing, which eliminates both <5>'s in R12C7.

The question is, can the logic of a finned xy-wing (either/or the xy-wing, the fin is true) be used for a finned xyz-wing or was this a lucky shot??

The "Either the wing ... OR the fin ..." constitutes a bifurcation like any other, (e.g. "either r1c3=4 or r1c3=9"), in my opinion no better or worse. If both assumptions lead to the same result, that result must be true.

When using such a "proof by exhaustion", you just got to be careful that no possible alternatives are forgotten. There must not be "a middle ground" (tertium no datur). Either case 1 or case 2 (or possibly both, but I prefer to make sure the cases don't overlap), but no third possibility.

My take on these finned creatures: Are finned xy(z) wings easier to spot than simple bi-value cells? I don't think so. Do they turn up a usable elimination more often than bi-value cells? Don't know but in general I doubt it. Personal conclusion: I dont go looking for them but sometimes use them when I step on one.

Finned x-wing are a slightly different matter (but still, this is strictly personal taste!) in that I usually have a sketch in front of me and both the x-wing pattern and the fin tend to stand out very clearly in those diagrams.

As for using the finned guys in order to obtain strong links not easily found otherwise, that is an idea with great potential value. With multi-coloring, w-wing and m-wing making extensive use of the simple strong links (those with only two "6" cells in a row for example), finding a strong link that jumps houses should be very valuable when solving the more difficult puzzles...

These "finned wings" function by creating a strong inference between the "fin" and the grouped pincer digits of the wing. It is not an "either/or" (conjugate) situation. It is possible that both are true. [In this finned XYZ Wing case, if the <7> fin is true, it is still possible that one of the two remaining <8> pincer digits is true; we don't know.] What is not possible is that both things are false.

It is obvious that if the "fin" is false, one or more of the pincer digits must be true since we then have a straightforward XYZ Wing. It is slightly less obvious that if all of the pincer digits are false then the fin must be true; however, that is the case. Once we accept this as given, then we only have to consider the strong inference that the "finned wing" contains. That means that we don't need to spend time thinking about "if this is true" or "if that is false" etc. We just construct an AIC that exploits the strong inference.

In Johan's example, the grouped pincer <8>s are: (8)r8c3|r9c18
The strong inference of the "finned wing" is: (7)r8c3=(8)r8c3|r9c18
To make it clear to others that you are working with XYZ Wing pincers, you could write:
(7)r8c3=XYZ Wing[(8)r8c3|r9c18]
It is then natural to add the weak link to one or more of the wing victims:
(7)r8c3=XYZ Wing[(8)r8c3|r9c18] - (8)r9c2

Once you have this core structure of the finned wing, you can append alternate links as you are comfortable with. Johan exploited the strong link on <8> in c2 plus bivalue sequences (as in XY chaining):

The <5>s on the ends of the chain are strongly linked so we know they can function as pincers for eliminations. If you like, you could also see this as a 58 W-Wing with the following external strong link on <8> that "activates" it:
(8=7)r5c3 - (7)r8c3=XYZ Wing[(8)r8c3|r9c18] - (8)r9c2=(8)r3c2

So, rather than thinking "true, false, true, false..." as I construct these sorts of things, I think "strong, weak, strong, weak..." as I trace out the links. It's like thinking "red, green, red, green..." when constructing a color chain (which I do without any marking in basic cases).

Since XY- and XYZ-Wings are just the simplest cases of the more general "paired ALS" eliminations, any such paired ALS "wing" can also be finned. And, W-Wings can be finned as well.

By the way, the "fin" doesn't need to be a single digit nor limited to a single cell of the wing! In principle, grouped digits can also serve as a "fin." However, you have to be certain that you can form a valid weak link to the grouped "fin" digits in order for this to work.

Asellus, thanks for the clarification! In my post, I was too quick to call the finned wing a "strong link" when in reality the strong link is only between the fin and the (grouped) pincers, as you pointed out.

A complete building block for eliminations would then have to include not only any (possibly grouped) victims

Quote:

It is then natural to add the weak link to one or more of the wing victims:
(7)r8c3=XYZ Wing[(8)r8c3|r9c18] - (8)r9c2

Once you have this core structure of the finned wing,

but also

Quote:

you can append alternate links as you are comfortable with

I would even go as far as saying that one more link is mandatory in all those cases where the fin does not "see" the victim directly. This addidional strong link provides that "other end" that the final victim will "see" (and get eliminated by seeing)

First, unless I'm missing something, I don't believe that the simple case is possible with finned XY- or XYZ-Wings. The fin would need to be the same digit as the victims/pincers. In the XYZ Wing case, all three cells already contain that digit (as pincers). There's no place for such a fin! In the XY Wing, the only place for such a fin is the pivot. But, that would just make it an XYZ Wing. Perhaps such an arrangement is possible in a 2-ALS "wing" that involves more cells, though I'm not certain (nor am I expecting to see an example any time soon!).

Second, I just want to make clear that one does not need to "weak link" to all of the (grouped) wing victims, though that is an option. The weak link can be to any single victim or to any (sub)set of the victims considered as a group... whatever works to provide the ensuing useful strong link. Often, it is only a "single victim" link that is useful.